1. EHSAN KHODDAM MOHAMMADI MILAD GANJALIZADEH BABAK YADEGARI First Steps to Study SCHEDULING بسم الله الرحمن الرحيم

2. DEFINITION allocation of scarce resources to activities with the objective of optimizing one or more performance measures

3. Example of RESOURCES assembly plant CPU memory and I/O devices runways at an airport mechanics in an automobile repair shop operations in a manufacturing process execution of a computer program landings and take-offs at an airport car repairs in an automobile repair shop Example of ACTIVITIES

4. SOME NECESSARY PARAMETERS (1/2) Number of jobs : n Number machines: m subscript j refers to a job subscript i refers to a machine Processing Time (pi j ): job j requires processing on machine i Release Date (r j ) : the time the job arrives at the system, which is the earliest time at which job j can start its processing. Due Date (dj ) : The due date dj of job j represents the date the job is expected to complete. Completion of a job after its due date is allowed, but it will incur a cost. Deadline (d¯ j ): The deadline d¯ j of job j represents the hard deadline that the job must respect Weight (w j ): The weight w j of job j reflects the importance of the job

5. SOME NECESSARY PARAMETERS (2/2) C j denote the completion time of job j. The lateness of job j is defined as: L j = C j − dj The tardiness of job j is defined as : Tj = max(L j, 0) The unit penalty of job j is defined as Uj = 1 if C j > dj otherwise, Uj = 0.

6. Introduction to Scheduling Scientific Notation PART1

7. NOTATION 3-field notation α|β|γ α : describes the machine environment and contains a single entry β : provides details of job characteristics and scheduling constraints. It may contain multiple entries or no entry γ : contains the objective function to optimize. It usually contains a single entry

8. α (Machine Field) Single Machine (1) : There is only one machine in the system. Parallel and Identical Machines (Pm): There are m identical machines in parallel Uniform Machines (Qm) : There are m machines in parallel, but the machines have different speeds Unrelated Machines (Rm ): There are m machines in parallel, but each machine can process the jobs in different paces Job Shop (Jm ): In a job shop with m machines, each job has its own predetermined route to follow. It may visit some machines more than once and it may not visit some machines at all Flow Shop (Fm ): In a flow shop with m machines, the machines are linearly ordered and the jobs all follow the same route (from the first machine to the last machine) Open Shop (Om ) : In an open shop with m machines, each job needs to be processed exactly once on each of the machines. But the order of processing is immaterial

9. β ( Job Characteristics ) Preemptions (pmtn) No-Wait (nwt) Precedence Constraints (prec) Release Dates (rj) Restrictions on the Number of Jobs (nbr ) Restrictions on the Number of Operations in Jobs (nj) Restrictions on the Processing Times (pj ) Deadlines (d ^ j )

10. γ (Objective Function) The objective to be minimized is always a function of the completion times of the jobs. Makespan (Cmax): The makespan is defined as max(C1,..., Cn). Maximum Lateness (Lmax):The maximum lateness is defined as max(L1,..., Ln). TotalWeighted Completion Time (wjC j ):The total (unweighted) completion time is denoted by C j. TotalWeighted Tardiness (w j Tj ):The total (unweighted) tardiness is denoted byTj. WeightedNumber ofTardy Jobs (wjUj ):The total (unweighted) number of tardy jobs is denoted byUj.

11. A Tutorial on Complexity PART2

12. Complexity Theory,WHY? Complexity theory is an important tool in scheduling research. When we are confronted with a new scheduling problem, the very first thing we try is to develop efficient algorithms for solving the problem. Unfortunately, very often, we could not come up with any algorithm more efficient than essentially an enumerative search, even though a considerable amount of time had been spent on the problem. In situations like this, the theory of NP-hardness may be useful to pinpoint that no efficient algorithms could possibly exist for the problem in hand. Therefore, knowledge of NP-hardness is absolutely essential for anyone interested in scheduling research.

13. P=NP ? (1/2) Class P : consists of those problems that are solvable in polynomial time Class NP : consists of those problems that are "verifiable" in polynomial time. What we mean here is that if we were somehow given a "certificate" of a solution, then we could verify that the certificate is correct in time polynomial in the size of the input to the problem. Class NPC (NP - complete) : if it is in NP and is as "hard" as any problem in NP

14. P=NP ? (2/2) if any NP-complete problem can be solved in polynomial time, then every NP-complete problem has a polynomial- time algorithm. If you can establish a problem as NP-complete, you provide good evidence for its intractability. As an engineer, you would then do better spending your time developing an approximation algorithm or solving a tractable special case (RELATION TO SCHEDULING)

15. Two main kind of problem (1/2) Decision Problem: the answer is simply "yes" or "no” Optimization Problem: each feasible (i.e., "legal") solution has an associated value, and we wish to find the feasible solution with the best value

16. Two main kind of problem (2/2) We usually can cast a given optimization problem as a related decision problem by imposing a bound on the value to be optimized. if a decision problem is hard, then related optimization problem is hard.

17. Partition Given a list A = (a 1, a 2,..., a n ) of n integers, can A be partitioned into A 1 and A 2 such that ? DECISION PROBLEM

18. 0/1-Knapsack Optimization Given a set U of n items, U = {u1, u2,..., un}, each item uj (sj=size, vj = value) a knapsack with size K Find a subset U ⊆ U such that all the items in U can be packed into the knapsack and such that the total value of the items in U is maximum.

19. 0/1-Knapsack Decision Given a set U of n items,U = {u1, u2,..., un} each item u j (s j =size, v j = value) a knapsack with size K and a bound B Is there a subset U` ⊆ U such that and ?

20. polynomial reducibility (1/2) Definition: Let P and Q be two decision problems. P is polynomially reducible Q ( P ∝ Q) 1. if there is a function f that maps every instance I P of P into an instance I Q of Q such that: I P is a Yes-instance I Q is a Yes-instance. 2. f can be computed in polynomial time.

22. polynomial reducibility (2/2) f does not have to be one-to-one the status of I P is unknown to f P ∝ Q does not imply that Q ∝ P reducibility is transitive: P ∝ Q, Q ∝ R P ∝ R.

23. Proof : Let A = (a 1, a 2,..., a n ) be a given instance of Partition. We create an instance of the decision version Of P 2 || C max as follows. Let there be n jobs, with job i having processing time a i. In essence, each job corresponds to an integer in A. Let the bound B be such. Clearly, the mapping can be done in polynomial time. It is easy to see that there is a partition of A if and only if there is a schedule with makespan no larger than B. ✷ Some SCHEDULING at last! Theorem : The Partition problem is reducible to the decision version of P2 || Cmax In the above reduction, 1.we create a job with processing time equal to an integer in the instance of the Partition problem. 2.The given integers can be partitioned into two equal groups if and only if the jobs can be scheduled on two parallel and identical machines with makespan equal to one half of the total processing time.

24. More SCHEDULING! Theorem : The 0/1-Knapsack Decision problem is reducible to the decision version of 1 | d j = d |. Proof : WHAT DO YOU THINK??? IF YOU CAN’T PROVE THIS, EMAIL TO ME ✷EMAIL TO ME HINT: In the above reduction, we create, for each item u j in the 0/1-Knapsack Decision, a job with a processing time equal to the size of u j and a weight equal to the value of u j. We make the knapsack size K to be the common due date of all the jobs. The idea is that if an item is packed into the knapsack, then the corresponding job is an on-time job; otherwise, it is a tardy job. Thus, there is a packing into the Knapsack with value greater than or equal to B if and only if there is a schedule with the total weight of all the tardy jobs less than or equal ✷ JUST KIDDING about E-mail! HINT IS ANSWER!

25. No More Scheduling!!! THANKS FOR YOUR ATTENTION

26. REFRENCES Handbook of Scheduling: Algorithms, Models and Performance Analysis, Joseph Y-T. Leung, CHAPMAN & HALL/CRC © 2004 Introduction to Algorithms, Second Edition by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest and Clifford Stein,The MIT Press © 2001